We consider a waveform relaxation method applied to the inhomogeneous heat equation with piecewise continuous initial and boundary conditions and a bounded H¨older continuous forcing function. Traditionally, to obtain a waveform relaxation algorithm, the equation is first discretized in space and the discrete matrix is split. Superlinear convergence on bounded time intervals can be shown but the error bounds are dependent on Lipschitz constants of the splitting which, in the case of the heat equation, typically blow up as x goes to zero. We split the partial differential equation (PDE) directly by using overlapping domain decomposition. We prove linear convergence of the algorithm in the continuous case on an infinite time interval, at a rate depending on the size of the overlap. This result remains valid after discretizing in space, leading to a waveform relaxation algorithm for the spatially discretized heat equation which exhibits linear convergence at a rate independent of the mesh parameter x. The algorithm is in the class of waveform relaxation algorithms based on over-lapping splittings. Numerical results are presented which support the convergence theory.